Lesson 8: Rotation Patterns

Let’s rotate figures in a plane.

8.1: Building a Quadrilateral

Here is a right isosceles triangle:

Right isosceles triangle A B C has horizonatl side A B with point A to the right of B, and has vertical side B C with point C directly above point B.
  1. Rotate triangle ABC 90 degrees clockwise around B
  2. Rotate triangle ABC 180 degrees clockwise round B.
  3. Rotate triangle ABC 270 degrees clockwise around B.
  4. What would it look like when you rotate the four triangles 90 degrees clockwise around B? 180 degrees? 270 degrees clockwise?

8.2: Rotating a Segment

Create a segment AB and a point C that is not on segment AB.

GeoGebra Applet YF2EDCTt

  1. Rotate segment AB 180^\circ around point B

  2. Rotate segment AB 180^\circ around point C

Construct the midpoint of segment AB with the Midpoint tool. 

  1. Rotate segment AB 180^\circ around its midpoint. What is the image of A?

  2. What happens when you rotate a segment 180^\circ?

8.3: A Pattern of Four Triangles

Here is a diagram built with three different rigid transformations of triangle ABC.

Use the applet to answer the questions. It may be helpful to reset the image after each question.

GeoGebra Applet Ccv3FucS

  1. Describe a rigid transformation that takes triangle ABC to triangle CDE.
  2. Describe a rigid transformation that takes triangle ABC to triangle EFG.
  3. Describe a rigid transformation that takes triangle ABC to triangle GHA.
  4. Do segments AC, CE, EG, and GA all have the same length? Explain your reasoning.

Summary

When we apply a 180-degree rotation to a line segment, there are several possible outcomes:

  • The segment maps to itself (if the center of rotation is the midpoint of the segment).
  • The image of the segment overlaps with the segment and lies on the same line (if the center of rotation is a point on the segment).
  • The image of the segment does not overlap with the segment (if the center of rotation is not on the segment).

We can also build patterns by rotating a shape. For example, triangle ABC shown here has m(\angle A) = 60. If we rotate triangle ABC 60 degrees, 120 degrees, 180 degrees, 240 degrees, and 300 degrees clockwise, we can build a hexagon.

Six identical equilateral triangles are drawn such that each triangle is aligned to another triangle created a hexagon. One of the triangle is labeled A B C and all 6 triangles meet at the common point of A.

Practice Problems ▶